Just casting about, I am seeing what seems like a fairly even distribution of interpretations of the Cyrillic lowercase shha (һ) U+04BB in fonts that support extended Cyrillic — some are basically a reduced, x-height form of the capital Shha (Һ), while others are a duplication of the Latin h.
This letter is part of several Cyrillic-based orthographies: some Turkic languages and some Mongolian ones.
What drives these two different forms? Are there some orthographies where an ascending glyph harmonizes better than in others? (Given the paucity of ascenders in Cyrillic script overall, the h form seems generally a less comfortable fit, to my eye.)
Are there local preferences in some traditions that lean toward one form over the other?